The document discusses Delaunay triangulation, which is a method for generating a triangulation of a set of points in a plane. It begins by motivating the problem of modeling a terrain from sample height points. It then discusses using triangulations to approximate the terrain, and defines properties of triangulations. It introduces the concept of Delaunay triangulation, which maximizes the minimum angle of all possible triangulations. It describes an algorithm called randomized incremental construction that computes the Delaunay triangulation by incrementally adding points and maintaining the Delaunay property through edge flips.
QGIS is a free and open-source cross-platform desktop geographic information system application that supports viewing, editing, and analysis of geospatial data.
QGIS is a free and open-source cross-platform desktop geographic information system application that supports viewing, editing, and analysis of geospatial data.
Plane and Applied surveying 2
Trigonometric Levelling Practical Part.
Report number(3)
Report name :
Apparatus
Theodolite instruments 1 No.
Range poles 1 No.
Tripod 2 Nos.
Surveyors’ pins 4 Nos.
Hammer 1 No.
Tape 1 No.
Object: The object is to measure the height of an inaccessible and accessible building using theodolite and measuring angles for the following cases:
1. Base of object is accessible.
2. Base of object is not accessible and the three points are on same vertical plane.
3. Base of object is not accessible and the three points are not on same vertical
plane.
-Procedure.
-Procedure for case 2.
-Procedure for case 3.
-Field Note Table.
Calculation For Case 1
I- If base of the object is accessible:
Calculation For Case 2
Case 2. The three points (A, B, and O) are on the same vertical plane.
-Calculations For case 3.
Asst. Prof. Salar K.Hussein
Mr. Kamal Y.Abdullah
Asst.Lecturer. Dilveen H. Omar
Erbil Polytechnic University
Technical Engineering College
Civil Engineering Department
Location. Location. Location. With so many maps and datums out there, how does a person know what datum is correct? How come my GPS coordinates don\'t match up on my map? Why is there a shift of 100 metres? How do I transform between different datums? What is a datum? What is the EPSG? Why have GIS Vendors and Oracle adopted them? Does offshore or onshore make a difference? How come there are so many datums? This presentation looks to provide some answers to some of these questions and to point out that latitude and longitude are not absolute.
Over the decades that surveyors have been trying to map the Earth, history and politics have shaped the way we see the world. Are the borders actually there? What if one nation adopts a standard, but the other does not? Does really matter what the co-ordinate system is? Why when I draw the a UTM Projection, the lines are curved, not in a grid? Is the OGC adopting these standards? So many questions and this presentation aims to answer some of them and provide some light on a complicated and sometimes unclear topic.
Comprehensive coverage of fundamentals of computer graphics.
3D Transformations
Reflections
3D Display methods
3D Object Representation
Polygon surfaces
Quadratic Surfaces
Photogrammetry Surveying, its Benefits & DrawbacksNI BT
Learn the Photogrammetry Surveying and benefits-drawbacks of photogrammetry. Photogrammetry is the process of generating a 3D model from a set of 2D photographs. In Surveying, this is done by taking two or more images of the same point from different angles
Plane and Applied surveying 2
Trigonometric Levelling Practical Part.
Report number(3)
Report name :
Apparatus
Theodolite instruments 1 No.
Range poles 1 No.
Tripod 2 Nos.
Surveyors’ pins 4 Nos.
Hammer 1 No.
Tape 1 No.
Object: The object is to measure the height of an inaccessible and accessible building using theodolite and measuring angles for the following cases:
1. Base of object is accessible.
2. Base of object is not accessible and the three points are on same vertical plane.
3. Base of object is not accessible and the three points are not on same vertical
plane.
-Procedure.
-Procedure for case 2.
-Procedure for case 3.
-Field Note Table.
Calculation For Case 1
I- If base of the object is accessible:
Calculation For Case 2
Case 2. The three points (A, B, and O) are on the same vertical plane.
-Calculations For case 3.
Asst. Prof. Salar K.Hussein
Mr. Kamal Y.Abdullah
Asst.Lecturer. Dilveen H. Omar
Erbil Polytechnic University
Technical Engineering College
Civil Engineering Department
Location. Location. Location. With so many maps and datums out there, how does a person know what datum is correct? How come my GPS coordinates don\'t match up on my map? Why is there a shift of 100 metres? How do I transform between different datums? What is a datum? What is the EPSG? Why have GIS Vendors and Oracle adopted them? Does offshore or onshore make a difference? How come there are so many datums? This presentation looks to provide some answers to some of these questions and to point out that latitude and longitude are not absolute.
Over the decades that surveyors have been trying to map the Earth, history and politics have shaped the way we see the world. Are the borders actually there? What if one nation adopts a standard, but the other does not? Does really matter what the co-ordinate system is? Why when I draw the a UTM Projection, the lines are curved, not in a grid? Is the OGC adopting these standards? So many questions and this presentation aims to answer some of them and provide some light on a complicated and sometimes unclear topic.
Comprehensive coverage of fundamentals of computer graphics.
3D Transformations
Reflections
3D Display methods
3D Object Representation
Polygon surfaces
Quadratic Surfaces
Photogrammetry Surveying, its Benefits & DrawbacksNI BT
Learn the Photogrammetry Surveying and benefits-drawbacks of photogrammetry. Photogrammetry is the process of generating a 3D model from a set of 2D photographs. In Surveying, this is done by taking two or more images of the same point from different angles
My talk about computational geometry in NTU's APEX Club in NTU, Singapore in 2007. The club is for people who are keen on participating in ACM International Collegiate Programming Contests organized by IBM annually.
Earliest Galaxies in the JADES Origins Field: Luminosity Function and Cosmic ...Sérgio Sacani
We characterize the earliest galaxy population in the JADES Origins Field (JOF), the deepest
imaging field observed with JWST. We make use of the ancillary Hubble optical images (5 filters
spanning 0.4−0.9µm) and novel JWST images with 14 filters spanning 0.8−5µm, including 7 mediumband filters, and reaching total exposure times of up to 46 hours per filter. We combine all our data
at > 2.3µm to construct an ultradeep image, reaching as deep as ≈ 31.4 AB mag in the stack and
30.3-31.0 AB mag (5σ, r = 0.1” circular aperture) in individual filters. We measure photometric
redshifts and use robust selection criteria to identify a sample of eight galaxy candidates at redshifts
z = 11.5 − 15. These objects show compact half-light radii of R1/2 ∼ 50 − 200pc, stellar masses of
M⋆ ∼ 107−108M⊙, and star-formation rates of SFR ∼ 0.1−1 M⊙ yr−1
. Our search finds no candidates
at 15 < z < 20, placing upper limits at these redshifts. We develop a forward modeling approach to
infer the properties of the evolving luminosity function without binning in redshift or luminosity that
marginalizes over the photometric redshift uncertainty of our candidate galaxies and incorporates the
impact of non-detections. We find a z = 12 luminosity function in good agreement with prior results,
and that the luminosity function normalization and UV luminosity density decline by a factor of ∼ 2.5
from z = 12 to z = 14. We discuss the possible implications of our results in the context of theoretical
models for evolution of the dark matter halo mass function.
(May 29th, 2024) Advancements in Intravital Microscopy- Insights for Preclini...Scintica Instrumentation
Intravital microscopy (IVM) is a powerful tool utilized to study cellular behavior over time and space in vivo. Much of our understanding of cell biology has been accomplished using various in vitro and ex vivo methods; however, these studies do not necessarily reflect the natural dynamics of biological processes. Unlike traditional cell culture or fixed tissue imaging, IVM allows for the ultra-fast high-resolution imaging of cellular processes over time and space and were studied in its natural environment. Real-time visualization of biological processes in the context of an intact organism helps maintain physiological relevance and provide insights into the progression of disease, response to treatments or developmental processes.
In this webinar we give an overview of advanced applications of the IVM system in preclinical research. IVIM technology is a provider of all-in-one intravital microscopy systems and solutions optimized for in vivo imaging of live animal models at sub-micron resolution. The system’s unique features and user-friendly software enables researchers to probe fast dynamic biological processes such as immune cell tracking, cell-cell interaction as well as vascularization and tumor metastasis with exceptional detail. This webinar will also give an overview of IVM being utilized in drug development, offering a view into the intricate interaction between drugs/nanoparticles and tissues in vivo and allows for the evaluation of therapeutic intervention in a variety of tissues and organs. This interdisciplinary collaboration continues to drive the advancements of novel therapeutic strategies.
Introduction:
RNA interference (RNAi) or Post-Transcriptional Gene Silencing (PTGS) is an important biological process for modulating eukaryotic gene expression.
It is highly conserved process of posttranscriptional gene silencing by which double stranded RNA (dsRNA) causes sequence-specific degradation of mRNA sequences.
dsRNA-induced gene silencing (RNAi) is reported in a wide range of eukaryotes ranging from worms, insects, mammals and plants.
This process mediates resistance to both endogenous parasitic and exogenous pathogenic nucleic acids, and regulates the expression of protein-coding genes.
What are small ncRNAs?
micro RNA (miRNA)
short interfering RNA (siRNA)
Properties of small non-coding RNA:
Involved in silencing mRNA transcripts.
Called “small” because they are usually only about 21-24 nucleotides long.
Synthesized by first cutting up longer precursor sequences (like the 61nt one that Lee discovered).
Silence an mRNA by base pairing with some sequence on the mRNA.
Discovery of siRNA?
The first small RNA:
In 1993 Rosalind Lee (Victor Ambros lab) was studying a non- coding gene in C. elegans, lin-4, that was involved in silencing of another gene, lin-14, at the appropriate time in the
development of the worm C. elegans.
Two small transcripts of lin-4 (22nt and 61nt) were found to be complementary to a sequence in the 3' UTR of lin-14.
Because lin-4 encoded no protein, she deduced that it must be these transcripts that are causing the silencing by RNA-RNA interactions.
Types of RNAi ( non coding RNA)
MiRNA
Length (23-25 nt)
Trans acting
Binds with target MRNA in mismatch
Translation inhibition
Si RNA
Length 21 nt.
Cis acting
Bind with target Mrna in perfect complementary sequence
Piwi-RNA
Length ; 25 to 36 nt.
Expressed in Germ Cells
Regulates trnasposomes activity
MECHANISM OF RNAI:
First the double-stranded RNA teams up with a protein complex named Dicer, which cuts the long RNA into short pieces.
Then another protein complex called RISC (RNA-induced silencing complex) discards one of the two RNA strands.
The RISC-docked, single-stranded RNA then pairs with the homologous mRNA and destroys it.
THE RISC COMPLEX:
RISC is large(>500kD) RNA multi- protein Binding complex which triggers MRNA degradation in response to MRNA
Unwinding of double stranded Si RNA by ATP independent Helicase
Active component of RISC is Ago proteins( ENDONUCLEASE) which cleave target MRNA.
DICER: endonuclease (RNase Family III)
Argonaute: Central Component of the RNA-Induced Silencing Complex (RISC)
One strand of the dsRNA produced by Dicer is retained in the RISC complex in association with Argonaute
ARGONAUTE PROTEIN :
1.PAZ(PIWI/Argonaute/ Zwille)- Recognition of target MRNA
2.PIWI (p-element induced wimpy Testis)- breaks Phosphodiester bond of mRNA.)RNAse H activity.
MiRNA:
The Double-stranded RNAs are naturally produced in eukaryotic cells during development, and they have a key role in regulating gene expression .
Multi-source connectivity as the driver of solar wind variability in the heli...Sérgio Sacani
The ambient solar wind that flls the heliosphere originates from multiple
sources in the solar corona and is highly structured. It is often described
as high-speed, relatively homogeneous, plasma streams from coronal
holes and slow-speed, highly variable, streams whose source regions are
under debate. A key goal of ESA/NASA’s Solar Orbiter mission is to identify
solar wind sources and understand what drives the complexity seen in the
heliosphere. By combining magnetic feld modelling and spectroscopic
techniques with high-resolution observations and measurements, we show
that the solar wind variability detected in situ by Solar Orbiter in March
2022 is driven by spatio-temporal changes in the magnetic connectivity to
multiple sources in the solar atmosphere. The magnetic feld footpoints
connected to the spacecraft moved from the boundaries of a coronal hole
to one active region (12961) and then across to another region (12957). This
is refected in the in situ measurements, which show the transition from fast
to highly Alfvénic then to slow solar wind that is disrupted by the arrival of
a coronal mass ejection. Our results describe solar wind variability at 0.5 au
but are applicable to near-Earth observatories.
Seminar of U.V. Spectroscopy by SAMIR PANDASAMIR PANDA
Spectroscopy is a branch of science dealing the study of interaction of electromagnetic radiation with matter.
Ultraviolet-visible spectroscopy refers to absorption spectroscopy or reflect spectroscopy in the UV-VIS spectral region.
Ultraviolet-visible spectroscopy is an analytical method that can measure the amount of light received by the analyte.
This pdf is about the Schizophrenia.
For more details visit on YouTube; @SELF-EXPLANATORY;
https://www.youtube.com/channel/UCAiarMZDNhe1A3Rnpr_WkzA/videos
Thanks...!
2. MOTIVATION: TERRAIN
TRIANGULATION
Property of triangulation
SOLUTION OF TERRAIN PROBLEME
ANGLE OPTIMAL TRIANGULATION
EDGE FLIPPING
THALES THEOREM
ILLEGAL TRIANGULATION
LEGAL TIANGULATION
DELAUNAY GRAPH
Property of Delaunay Graph
DELAUNAY TRIANGULATION
Property of Delaunay Triangulation
Computing Delaunay Triangulation
RANDOMIZED INCREMENTAL CONSTRUCTION
Application of Delaunay Triangulation
Euclidean Minimum Spanning Tree
Euclidean Traveling salesperson Problem
REFERENCES
3. • We can model a piece of the earth’s surface as a terrain.
• A terrain is a 2-dimensional surface in 3-dimensional space with a
special property: every vertical line intersects it in a point, if it
intersects it at all. A terrain can be visualized with a perspective
drawing like the one in bellows Figure
Figure: A perspective view of Terrain
4. • To build a model of the terrain surface, we can start with a
number of sample points where we know the height.
• Of course, we don’t know the height of every point on earth; we only
know it where we’ve measured it.
• This means that when we talk about some terrain, we only know the
value of the function f at a finite set P ⊂ A of sample points.
• From the height of the sample points we somehow have to
approximate the height at the other points in the domain.
• A native approach assigns to every p ∈ A the height of the nearest
sample point. However, this gives a discrete terrain, which doesn’t
look very natural.
Figure: A terrain from a set of sample points
5. So, How can we most naturally approximate height
of points not in A?
• Determine a
Triangulation of A in
R², then raise points to
desired height
• Triangulation: planar
subdivision whose
bounded faces are
triangles with vertices
from A
Figure : Triangulation
6. Maximal planar subdivision: a subdivision S such that
no edge connecting two vertices can be added to S without
destroying its planarity
Triangulation of set of points P: a maximal planar
subdivision whose vertices are elements of P.
Plane Graph: No two edges in the embedding cross.
7. • Outer polygon must be
convex hull that is the
union of the bounded faces
of T is always the convex
hull of P, and that the
unbounded face is always
the complement of the
convex hull.
• Internal faces must be
triangles, otherwise they
could be triangulated
further. Convex Hull
8. Here, Some triangles are better than other in triangulation.
SO, now remain a question that-
But which triangulation is the most appropriate one for
our purpose, namely to approximate a terrain??
• There is no definitive answer to this question. We do not know the
original terrain, we only know its height at the sample points.
Since we have no other information, and the height at the sample
points is the correct height for any triangulation, all triangulations
of P seem equally good. Nevertheless, some triangulations look
more natural than others.
• So here we will avoid skinny triangles. That means we will try
maximize the minimum angles which is basically DELAUNAY
TRIANGULATION
SO, TERRAIN Problem solve by Angle Optimal
Triangulation(Edge Flipping) and Delaunay Triangulation
9. To create a angle vector Let T be a triangulation of P with m
triangles. Its angle vector is A(T) = (α₁,α₂,α₃,.....,α₃m) where
α₁,α₂,α₃,……,α₃m are the angles of T sorted by increasing
value.
Let Tʹ is another triangulation of P. Its angle vector is A(Tʹ)
= (αʹ₁,αʹ₂,αʹ₃,.......,αʹ₃m) Then we say the angle-vector of T is
larger than the angle-vector of Tʹ if A(T) is lexicographically
larger than A(Tʹ). This means iff there exists an index i with
1≤ i ≤3m such that αj=αjʹ for all j<i and αᵢ>αᵢʹ then
A(T)> A(Tʹ).
So , Best triangulation is triangulation that is angle optimal,
i.e. has the largest angle vector. That is Maximizes
minimum angle. Then we say the triangulation T is angle-
optimal if A(T) ≥ A(Tʹ).
.
10. • If the two triangles form a convex quadrilateral, we could have an
alternative triangulation by performing an Edge Flip on their shared
edge.
• In the Edge Flipping just change diagonal or an edge which shared
by two triangles and also change angles.
Edge Flip
• Change the angle vector (α₁,……,α₆) replaced by αʹ₁,……,αʹ₆)
• The edge e = PiPj is Illegal Edge if min αᵢ 1≤i≤6 < min α’ᵢ 1≤i≤6
• Flipping an illegal edge increases the angle vector
11. • We can use Thale’s Theorem to test if an edge is legal
without calculating angles
Denote the smaller angle
defined by three points p, q,
r.
Let C be a circle, l a line
intersecting C in points a and
b and p, q, r, and s points
lying on the same side of l.
Suppose that p and q lie on C,
that r lies inside C, and that s
lies outside C. Then:
12. • If triangulation T contains an illegal edge e, we can
make A(T) larger by flipping e.
• In this case, T is an illegal triangulation.
• The edge PiPj is illegal if and only if Pl lies in the
interior of the circle C.
Pl
Illegal Edge
C
13. A legal triangulation is a triangulation that does not contain
any illegal edge.
Algorithm LegalTriangulation(T)
Input. A triangulation T of a point set P.
Output. A legal triangulation of P.
1. while T contains an illegal edge pipj
2. do ( Flip PiPj )
3. Let PiPjPk and PiPjPl be the two triangles adjacent to pipj.
4. Remove PiPj from T, and add PkPl instead.
5. return T
Flip illegal edges of this triangulation until all edges are legal . Algorithm
terminates because there is a finite number of triangulations of input points
P.
14. The strait line embedding of a graph G, which has a line
connecting two nodes if their corresponding Voronoi cells
share an edge, denoted as Delaunay Graph of P ie. DG(P).
DG(P)
15. • No two edges cross; DG(P) is a planar graph.
PiPj is in DG(P) iff ∃ closed disc Cij
with Pi and Pjon the boundary and no
other site is contained in it.
Let tij be the triangle formed by Pi,Pj
and center of Cij.
Note that edge of tij between Pi and
center of Cij is inside V(Pi).
Let another edge PkPl also in DG(P)
such that PiPj and PkPl
intersect.Define Ckl and tkl for PkPl.If
intersected PiPj, it must also intersect
one other edge e of tij.Because Pk Contained in v(Pi)
Contained in v(Pi)
and Pl are outside Cij and therefore outside tij.Say an edge e’ of tkl must
intersect PiPj.
Notice that one of the edges of tij incident to center of Cij and one of the edges
of tkl incident to center of Ckl must intersect.
But, those edges must be contained within their respective Voronoi cells,
which is a contradiction.
16. If the point set P is in general position then the Delaunay
graph is a triangulation.
General Position: We say that a set of points in general position if it
contains no four points on a circle. Therefore, P in general position means
every vertex of the Voronoi diagram has degree three, and consequently
all bounded faces of DG(P) are triangles.
Delaunay Triangulation
DG(P)
VOR(P)
17. Let P be a set of points in the plane, and let T be a triangulation of P. Then
T is a Delaunay triangulation of P if and only if the circumcircle of any
triangle of T does not contain a point of P in its interior.If the circumcircle
of any triangle does not contain other points of P then its called Empty
Circle property.
18. Let P be a set of points in the plane. A triangulation T of P is
legal if and only if T is a Delaunay triangulation of P.
We shall prove that any legal triangulation is
Delaunay Triangulation by contradiction.So assume
T is a legal triangulation of P that is not Delaunay
Triangulation.As T is legal so adjacent edge of
triangles is legal.As T is not Delaunay Triangulation
so there is a ΔPiPjPk ,the circum circle C(PiPjPk)
contains a point Pl ϵ P in its interior. Let e=PiPj be
the edge of pipj pl such that the triangle PiPjPl does
not intersect PiPjPk Now look at the triangle pipj pm
adjacent to pipj pk along e. Since T is legal, e is
legal.so Pm does not lie in the interior circle of
Pm
Pk
Pj
Pi
Pl
C(PiPjPk).The citcum circle C(PiPjPm) of ΔPiPjPm contains the part of C(PiPjPk), separeted
from PiPjPk by e.Consequently Pl ϵ C(PiPjPk).Assume that edge PjPm is the edge of the
ΔPiPjPl so this is not intersect ΔPiPjPm.But by Thales Theorem, ∠PjPlPi > ∠PjPmPi.
a contradiction as it allows PjPm to be flipped.
19. Let P be a set of points in the plane. Any angle-optimal triangulation of P
is a Delaunay Triangulation of P. That is any Delaunay triangulation of P
maximizes the minimum angle over all triangulations of P.
A
B
C
D
90°
A
B
C
D
min αᵢ = 27° min αᵢ = 50°After edge flip for
angle optimal
triangulation
20. • If we can verify for every edges that the two triangles are sharing the
edges are legal then it if Delaunay Triangulation.
• We can’t take every circum circle and check the respectively point.
But we only look at the edge which is sharing the two triangles and
draw the circum circles if they are not contain any other point and
this is holds for every pair of trianngles then it’s a Delaunay
Triangulation. This is called Local Property
21. 1. This algorithm start with a big triangle say P-1P-2P-3 that contains all the points.
The vertices of this big triangle should not be lie in any circle defined by 3
points in P.
2. Add one point Pr at a time, then add edges from Pr to the verticies of the
existing triangle.
3. Then two cases arise here:
Pr lies in the interior of the triangle. (figure: A)
Pr falls on an edge (need to make sure new edges are legal by flipping
edges if necessary). (figure B)
Pi
Pj
Pk
Pr
Pk
Pj
Pl
Pi
Pr
A simple algorithm to compute Delaunay Trianguletion. It is Randomize
Incremental Construction
22. Input: A set P of n points in the plane
Output: A Delaunay triangulation of P
1. Let p-1 , p-2 & p-3 be 3 point s.t. P is contained in triangle p-1 p-2 p-3
2. Initialize T as triangulation consisting of a single triangle p-1 p-2 p-3
3. Compute a random permutation p1 , … , pn of P
4. for r 1 to n
5. do (* Insert pr into T : *)
6. Find a triangle pi pj pk T containing pr
7. if pr lies in the interior of the triangle pi pj pk
8. then Add edges from pr to 3 vertices of pi pj pk and split it
9. LegalizeEdge(pr , pi pj , T)
10. LegalizeEdge(pr , pj pk , T)
11. LegalizeEdge(pr , pk pi , T)
23. 12. else (* pr lies on an edge of pi pj pk, say the edge pi pj *)
13. Add edges from pr to pk and to the third vertex of
other triangle that is incident to pi pj , thereby
splitting 2 triangles incident to pi pj into 4 tri’s
14. LegalizeEdge(pr , pi pl , T)
15. LegalizeEdge(pr , pl pj , T)
16. LegalizeEdge(pr , pj pk , T)
17. LegalizeEdge(pr , pk pi , T)
18. Discard p-1 , p-2 and p-3 with all their incident edges from T
19. return T
24. Algorithm_Legalize_Edge(pr , pi pj , T)
1. (* The point being inserted is pr , and pi pj is the edge of T
that may need to be flipped *)
2. if pi pj is illegal
3. then Let pi pj pk be triangle adjacent to pr pi pj along pi pj
4. (* Flip pi pj *) Replace pi pj with pr pk
5. LegalizeEdge(pr , pi pk , T)
6. LegalizeEdge(pr , pk pj , T)
25. Pi
Pr
Pk
Pj
This meet the Delaunay Triangulation
Here after adding this Pr, the circum circle of this intirior point contain
some other point of set P.So,Now flipping the edge for restore the
Delaunay Property.
But,
Pr
26. After flipping we get new two tryangle
and draw the circum circle and check
the Delaunay Property,but this figure
again not meet the Delaunay
Property.So we flipping the edge again.
P
r
27. After flipping edge we check again
Delaunay Property and this
triangulation meet the Delaunay
property because the ciecum circle is not
contain ny othe point. So, Now this
triangulation is Delaunay
Triangulation.
This flipping is continue until the triangulation becomes legal
and every time flip only the diagonal one of the end point of the
diagonal is the new point and then the previous triangles are
legal so it must be include the new point
P
r
28. In the Algorithm of Randomize Incremental Construction , find the
triangle which containing Pr point.
While we build the Delaunay triangulation, we also build a point
location structure D. The leaves of D correspond to the triangles of
current triangulation T. We maintain cross pointers between the leaves
and the triangulation.
Each node in D corresponds to triangle that was created by some point.
We initialize D as a DAG(Directed Acyclic Graph) with a single leaf
node that corresponds to the big triangle.
Searching for a point Pr would be going down the DAG D through a
sequence of nodes corresponding to all triangles created before step r
that contain Pr.
Whenever we split a triangle 1 into smaller triangles a and b (and
possibly c), add the smaller triangles to D as leaves of 1
For the rest, see the pictorial example.
31. Let M be the maximum value of any
coordinate of a point in P. Then , big
the triangle has the vertices at
P-1 = (3M, 0),
P-2 = (0, 3M)
and
P-3 = (- 3M, - 3M).
This are not actual coordinate.
M
M
32. The Randomized Incremental Algorithm for the Delaunay
Triangulation of n points takes O(n log n) time and O(n)
space, both on expectation.
From nodes in D representing triangles created .Each triangle takes O(1) time, so
the total time will be O(n) + (time for point locations).
Now Searching for a point Pr would entail going down the DAG D through a
sequence of nodes corresponding to all triangles created before step r that
contain Pr.
So each triangle that was created is visited once for each point in it. In other
words, . . . Let K(∆) ⊆ P be the points inside triangle ∆. Time for point locations is
∆
K(∆)
each triangle created in round r has O(n/r) points of P in it. Therefore,
∆
K(∆) ≤
𝒓
O(n/r)
ϵ O n(log n)
33. Euclidean Minimum Spanning Trees
For a set P of n points in the plane, the
Euclidean Minimum Spanning Tree is the
graph with minimum summed edge length
that connects all points in P and has only
the points of P as vertices
The Euclidean Minimum
Spanning Tree does not have
cycles.
Suppose G is the shortest connected
graph and it has a cycle. Removing one
edge from the cycle makes a new graph
G0 that is still connected but which is
shorter. Contradiction
Euclidean Minimum
Spanning Tree
34. Every edge of the Euclidean Minimum Spanning Tree is an
edge in the Delaunay graph
• Suppose T is an EMST with an
edge e = pq that is not Delaunay
Consider the circle C that has e as
its diameter. Since e is not
Delaunay, C must contain another
point r in P.
• Either the path in T from r to p
passes through q, or vice versa.
The cases are symmetric, so we
can assume the former case.
• Then removing e and inserting pr
instead will give a connected
graph again (in fact, a tree)
• Since q was the furthest point
from p inside C, r is closer to q, so
T was not a minimum spanning
tree. Contradiction
P
q
r
e
T
Which is not
possible
T’
35. A data structure exists that maintains disjoint sets and allows the following two
operations:
Union: Takes two sets and makes one new set that is the union.
Find: Takes one element and returns the name of the set that contains it
If there are n elements in total, then all Unions together take O(nlogn) time and each
Find operation takes O(1) time
Let P be a set of n points in the plane for which we want to compute the
EMST
1. Make a Union-Find structure where every point of P is in a separate set
2. Construct the Delaunay triangulation DT of P
3. Take all edges of DT and sort them by length
4. For all edges e from short to long:
Let the endpoints of e be p and q
If Find(p) ≠ Find(q), then put e in the EMST, and
Union(Find(p),Find(q))
So, The Euclidean Minimum Spanning Tree of P can be computed in O(nlogn)
time
36. • Given a set P of n points in the plane, the Euclidean Traveling
Salesperson Problem is to compute a tour (cycle) that visits all points of P
and has minimum length . A tour is an order on the points of P (cyclic
order). A set of n points has (n-1)! different tours
• We can determine the length of each tour in O(n) time: a brute-force
algorithm to solve the Euclidean Traveling Salesperson Problem (ETSP)
takes O(n) O((n-1)!) = O(n!) time
• If an algorithm A solves an optimization problem always within a factor
k of the optimum, then A is called an k-approximation algorithm.
• If an instance I of ETSP has an optimal solution of length L, then a k-
approximation algorithm will and a tour of length ≤ k L
37. We will use the EMST to approximate the ETSP
start at any vertexfollow an edge on one sideto get to another vertexproceed this wayskipping visited verticesand close the tour
38. • The walk visits every edge twice,
so it has length 2│EMST│
• The tour skips vertices, which
means the tour has length ≤
2│EMST│
• The optimal ETSP-tour is a
spanning tree if you remove any
edge!!!
So │EMST│ < │ETSP│
• Remove the edge PiPj . So now this
ETSP is EMST.
Pi
Pj
A tour visiting all points whose length is at most twice the
minimum possible can be computed in O(nlogn) time. So an
O(nlogn) time, 2-approximation for ETSP exists
As ETMST is meet the Delaunay property and evry edges are Delaunay
edges then ETSP is also meet Delaunay Property and its edges are
Delaunay edges